Page 1
DIFFERENTIAL EQUATION
Differential equations are a cornerstone of advanced mathematics, bridging the gap between
calculus and realworld applications. These equations describe relationships involving functions
and their derivatives, allowing us to model complex phenomena in physics, engineering,
economics, and many other fields.
This section introduces the fundamental concepts of differential equations, focusing on ordinary
differential equations. We'll explore the definition, order, and degree of these equations, as well as
methods for forming and solving them. By understanding differential equations, we gain powerful
tools for analyzing dynamic systems and predicting how quantities change over time.
From basic classifications to solution techniques, this material provides a foundation for working
with differential equations, setting the stage for more advanced study and practical applications in
various scientific disciplines.
1. DIFFERENTIAL EQUATION :
An equation that involves independent and dependent variables and the derivatives of the
dependent variables is called a differential equation.
A differential equation is said to be ordinary, if the differential coefficients have reference to a
single independent variable only e.g.
?? 2
?? ????
2

2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are
two or more independent variables. e.g.
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are
concerned with ordinary differential equations only.
2. ORDER OF DIFFERENTIAL EQUATION :
The order of a differential equation is the order of the highest differential coefficient occurring in
it.
3. DEGREE OF DIFFERENTIAL EQUATION:
The exponent of the highest order differential coefficient, when the differential equation is
expressed as a polynomial in all the differential coefficient.
Thus the differential equation :
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ??  1
( ?? )
?? ?? ??  1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ??
Note :
(i) The exponents of all the differential coefficient should be free from radicals and fraction.
(ii) The degree is always positive natural number.
(iii) The degree of differential equation may or may not exist.
Page 2
DIFFERENTIAL EQUATION
Differential equations are a cornerstone of advanced mathematics, bridging the gap between
calculus and realworld applications. These equations describe relationships involving functions
and their derivatives, allowing us to model complex phenomena in physics, engineering,
economics, and many other fields.
This section introduces the fundamental concepts of differential equations, focusing on ordinary
differential equations. We'll explore the definition, order, and degree of these equations, as well as
methods for forming and solving them. By understanding differential equations, we gain powerful
tools for analyzing dynamic systems and predicting how quantities change over time.
From basic classifications to solution techniques, this material provides a foundation for working
with differential equations, setting the stage for more advanced study and practical applications in
various scientific disciplines.
1. DIFFERENTIAL EQUATION :
An equation that involves independent and dependent variables and the derivatives of the
dependent variables is called a differential equation.
A differential equation is said to be ordinary, if the differential coefficients have reference to a
single independent variable only e.g.
?? 2
?? ????
2

2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are
two or more independent variables. e.g.
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are
concerned with ordinary differential equations only.
2. ORDER OF DIFFERENTIAL EQUATION :
The order of a differential equation is the order of the highest differential coefficient occurring in
it.
3. DEGREE OF DIFFERENTIAL EQUATION:
The exponent of the highest order differential coefficient, when the differential equation is
expressed as a polynomial in all the differential coefficient.
Thus the differential equation :
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ??  1
( ?? )
?? ?? ??  1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ??
Note :
(i) The exponents of all the differential coefficient should be free from radicals and fraction.
(ii) The degree is always positive natural number.
(iii) The degree of differential equation may or may not exist.
Problem 1 : Find the order and degree of the following differential equation :
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
(ii)
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
)
(iii)
????
????
= v 3 ?? + 5
Solution :
(i) The given differential equation can be rewritten as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
Hence order is 2 and degree is 3 .
(ii) The given differential equation has the order 2. Since the given differential equation cannot be
written as a polynomial in the differential coefficients, the degree of the equation is not defined.
(iii) Its order is 1 and degree 1.
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are 
(A) 2,2
(B) 2 , 3
(C) 3,2
(D) none of these
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential
equations).
Ans. (A)
4. FORMATION OF A DIFFERENTIAL
EQUATION :
In order to obtain a differential equation whose solution is
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which
we require ( ?? + 1 ) equations.
A differential equation is obtained as follows :
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the
number of independent arbitrary constants in it.
(b) Eliminate the arbitrary constants.
(c) The eliminant is the required differential equation.
Page 3
DIFFERENTIAL EQUATION
Differential equations are a cornerstone of advanced mathematics, bridging the gap between
calculus and realworld applications. These equations describe relationships involving functions
and their derivatives, allowing us to model complex phenomena in physics, engineering,
economics, and many other fields.
This section introduces the fundamental concepts of differential equations, focusing on ordinary
differential equations. We'll explore the definition, order, and degree of these equations, as well as
methods for forming and solving them. By understanding differential equations, we gain powerful
tools for analyzing dynamic systems and predicting how quantities change over time.
From basic classifications to solution techniques, this material provides a foundation for working
with differential equations, setting the stage for more advanced study and practical applications in
various scientific disciplines.
1. DIFFERENTIAL EQUATION :
An equation that involves independent and dependent variables and the derivatives of the
dependent variables is called a differential equation.
A differential equation is said to be ordinary, if the differential coefficients have reference to a
single independent variable only e.g.
?? 2
?? ????
2

2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are
two or more independent variables. e.g.
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are
concerned with ordinary differential equations only.
2. ORDER OF DIFFERENTIAL EQUATION :
The order of a differential equation is the order of the highest differential coefficient occurring in
it.
3. DEGREE OF DIFFERENTIAL EQUATION:
The exponent of the highest order differential coefficient, when the differential equation is
expressed as a polynomial in all the differential coefficient.
Thus the differential equation :
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ??  1
( ?? )
?? ?? ??  1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ??
Note :
(i) The exponents of all the differential coefficient should be free from radicals and fraction.
(ii) The degree is always positive natural number.
(iii) The degree of differential equation may or may not exist.
Problem 1 : Find the order and degree of the following differential equation :
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
(ii)
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
)
(iii)
????
????
= v 3 ?? + 5
Solution :
(i) The given differential equation can be rewritten as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
Hence order is 2 and degree is 3 .
(ii) The given differential equation has the order 2. Since the given differential equation cannot be
written as a polynomial in the differential coefficients, the degree of the equation is not defined.
(iii) Its order is 1 and degree 1.
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are 
(A) 2,2
(B) 2 , 3
(C) 3,2
(D) none of these
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential
equations).
Ans. (A)
4. FORMATION OF A DIFFERENTIAL
EQUATION :
In order to obtain a differential equation whose solution is
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which
we require ( ?? + 1 ) equations.
A differential equation is obtained as follows :
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the
number of independent arbitrary constants in it.
(b) Eliminate the arbitrary constants.
(c) The eliminant is the required differential equation.
Note :
(i) A differential equation represents a family of curves all satisfying some common properties.
This can be considered as the geometrical interpretation of the differential equation.
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h
order
i.e. equal to number of independent arbitrary constant.
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? axis and
having latus rectum ?? .
Solution : Equation of parabola whose axes is parallel to ?? axis and having latus rectum 'a' is ( ?? 
?? )
2
= ?? ( ??  ?? )
Differentiating both sides, we get 2 ( ??  ?? )
????
????
= ??
Again differentiating, we get
? 2 ( ??  ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
Writing the value of c from (i), we have
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
i.e.
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ??
Hence ?? =
2 ?? 3
[
????
????
]  ??
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
)  ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
,
which is the required differential equation.
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ??
?? ?? ?? ? ???? , where ?? = fixed constant
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant
Differentiating, we get
????
????
=  ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ?
Page 4
DIFFERENTIAL EQUATION
Differential equations are a cornerstone of advanced mathematics, bridging the gap between
calculus and realworld applications. These equations describe relationships involving functions
and their derivatives, allowing us to model complex phenomena in physics, engineering,
economics, and many other fields.
This section introduces the fundamental concepts of differential equations, focusing on ordinary
differential equations. We'll explore the definition, order, and degree of these equations, as well as
methods for forming and solving them. By understanding differential equations, we gain powerful
tools for analyzing dynamic systems and predicting how quantities change over time.
From basic classifications to solution techniques, this material provides a foundation for working
with differential equations, setting the stage for more advanced study and practical applications in
various scientific disciplines.
1. DIFFERENTIAL EQUATION :
An equation that involves independent and dependent variables and the derivatives of the
dependent variables is called a differential equation.
A differential equation is said to be ordinary, if the differential coefficients have reference to a
single independent variable only e.g.
?? 2
?? ????
2

2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are
two or more independent variables. e.g.
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are
concerned with ordinary differential equations only.
2. ORDER OF DIFFERENTIAL EQUATION :
The order of a differential equation is the order of the highest differential coefficient occurring in
it.
3. DEGREE OF DIFFERENTIAL EQUATION:
The exponent of the highest order differential coefficient, when the differential equation is
expressed as a polynomial in all the differential coefficient.
Thus the differential equation :
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ??  1
( ?? )
?? ?? ??  1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ??
Note :
(i) The exponents of all the differential coefficient should be free from radicals and fraction.
(ii) The degree is always positive natural number.
(iii) The degree of differential equation may or may not exist.
Problem 1 : Find the order and degree of the following differential equation :
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
(ii)
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
)
(iii)
????
????
= v 3 ?? + 5
Solution :
(i) The given differential equation can be rewritten as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
Hence order is 2 and degree is 3 .
(ii) The given differential equation has the order 2. Since the given differential equation cannot be
written as a polynomial in the differential coefficients, the degree of the equation is not defined.
(iii) Its order is 1 and degree 1.
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are 
(A) 2,2
(B) 2 , 3
(C) 3,2
(D) none of these
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential
equations).
Ans. (A)
4. FORMATION OF A DIFFERENTIAL
EQUATION :
In order to obtain a differential equation whose solution is
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which
we require ( ?? + 1 ) equations.
A differential equation is obtained as follows :
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the
number of independent arbitrary constants in it.
(b) Eliminate the arbitrary constants.
(c) The eliminant is the required differential equation.
Note :
(i) A differential equation represents a family of curves all satisfying some common properties.
This can be considered as the geometrical interpretation of the differential equation.
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h
order
i.e. equal to number of independent arbitrary constant.
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? axis and
having latus rectum ?? .
Solution : Equation of parabola whose axes is parallel to ?? axis and having latus rectum 'a' is ( ?? 
?? )
2
= ?? ( ??  ?? )
Differentiating both sides, we get 2 ( ??  ?? )
????
????
= ??
Again differentiating, we get
? 2 ( ??  ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
Writing the value of c from (i), we have
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
i.e.
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ??
Hence ?? =
2 ?? 3
[
????
????
]  ??
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
)  ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
,
which is the required differential equation.
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ??
?? ?? ?? ? ???? , where ?? = fixed constant
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant
Differentiating, we get
????
????
=  ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ?
Again differentiating, we get
?? 2
?? ?? ?? 2
=  ?? 2
?? ?? ?? ?? ? ????  ?? 2
?? ?? ?? ?? ? ????
using equation (i), we get ?
?? 2
?? ?? ?? 2
=  ?? 2
??
Do yourself  2
Eliminate the arbitrary constants and obtain the differential equation satisfied by it.
(i) ?? = 2 ?? + ????
??
(ii) ?? = (
?? ?? 2
) + ????
(iii) ?? = ?? ?? 2 ?? + ?? ??  2 ?? + ??
5. SOLUTION OF DIFFERENTIAL
EQUATION :
The solution of the differential equation is a relation between the variables of the equation not
containing the derivatives, but satisfying the given differential equation (i.e., from which the given
differential equation can be derived).
Thus, the solution of
????
????
= ?? ?? could be obtained by simply integrating both sides, i.e., ?? = ?? ?? + ??
and that of,
????
????
= ???? + ?? is ?? =
????
2
2
+ ???? + ?? , where ?? is arbitrary constant.
(i) A general solution or an integral of a differential equation is a relation between the variables
(not involving the derivatives) which contains the same number of the arbitrary constants as the
order of the differential equation.
For example, a general solution of the differential equation
?? 2
?? ?? ?? 2
=  4 ?? is ?? = ???? ?? ?? ? 2 ?? + ?? ?? ?? ?? ? 2 ??
where ?? and ?? are the arbitrary constants.
(ii) Particular solution or particular integral is that solution of the differential equation which is
obtained from the general solution by assigning particular values to the arbitrary constant in the
general solution.
For example, ?? = 10 ?? ?? ?? ? 2 ?? + 5 ?? ?? ?? ? 2 ?? is a particular solution of differential equation
?? 2
?? ????
2
=  4 ?? .
Note :
(i) The general solution of a differential equation can be expressed in different (but equivalent)
forms. For example
?????? ? ??  ?????? ? ( ?? + 2 ) = ?? # ( ?? ) ?
where ?? is an arbitrary constant is the general solution of the differential equation ????
'
= ?? + 2.
The solution given by equation (i) can also be rewritten as
?????? ? (
?? ?? + 2
) = ?? ????
?? ?? + 2
= ?? ?? = ?? 1
# ( ???? ) ? ???? ? ?? = ?? 1
( ?? + 2 ) # ( ?????? ) ?
where ?? 1
= ?? ?? is another arbitrary constant. The solution (iii) can also be written as
Page 5
DIFFERENTIAL EQUATION
Differential equations are a cornerstone of advanced mathematics, bridging the gap between
calculus and realworld applications. These equations describe relationships involving functions
and their derivatives, allowing us to model complex phenomena in physics, engineering,
economics, and many other fields.
This section introduces the fundamental concepts of differential equations, focusing on ordinary
differential equations. We'll explore the definition, order, and degree of these equations, as well as
methods for forming and solving them. By understanding differential equations, we gain powerful
tools for analyzing dynamic systems and predicting how quantities change over time.
From basic classifications to solution techniques, this material provides a foundation for working
with differential equations, setting the stage for more advanced study and practical applications in
various scientific disciplines.
1. DIFFERENTIAL EQUATION :
An equation that involves independent and dependent variables and the derivatives of the
dependent variables is called a differential equation.
A differential equation is said to be ordinary, if the differential coefficients have reference to a
single independent variable only e.g.
?? 2
?? ????
2

2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are
two or more independent variables. e.g.
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are
concerned with ordinary differential equations only.
2. ORDER OF DIFFERENTIAL EQUATION :
The order of a differential equation is the order of the highest differential coefficient occurring in
it.
3. DEGREE OF DIFFERENTIAL EQUATION:
The exponent of the highest order differential coefficient, when the differential equation is
expressed as a polynomial in all the differential coefficient.
Thus the differential equation :
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ??  1
( ?? )
?? ?? ??  1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ??
Note :
(i) The exponents of all the differential coefficient should be free from radicals and fraction.
(ii) The degree is always positive natural number.
(iii) The degree of differential equation may or may not exist.
Problem 1 : Find the order and degree of the following differential equation :
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
(ii)
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
)
(iii)
????
????
= v 3 ?? + 5
Solution :
(i) The given differential equation can be rewritten as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
Hence order is 2 and degree is 3 .
(ii) The given differential equation has the order 2. Since the given differential equation cannot be
written as a polynomial in the differential coefficients, the degree of the equation is not defined.
(iii) Its order is 1 and degree 1.
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are 
(A) 2,2
(B) 2 , 3
(C) 3,2
(D) none of these
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential
equations).
Ans. (A)
4. FORMATION OF A DIFFERENTIAL
EQUATION :
In order to obtain a differential equation whose solution is
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which
we require ( ?? + 1 ) equations.
A differential equation is obtained as follows :
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the
number of independent arbitrary constants in it.
(b) Eliminate the arbitrary constants.
(c) The eliminant is the required differential equation.
Note :
(i) A differential equation represents a family of curves all satisfying some common properties.
This can be considered as the geometrical interpretation of the differential equation.
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h
order
i.e. equal to number of independent arbitrary constant.
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? axis and
having latus rectum ?? .
Solution : Equation of parabola whose axes is parallel to ?? axis and having latus rectum 'a' is ( ?? 
?? )
2
= ?? ( ??  ?? )
Differentiating both sides, we get 2 ( ??  ?? )
????
????
= ??
Again differentiating, we get
? 2 ( ??  ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
Writing the value of c from (i), we have
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
i.e.
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ??
Hence ?? =
2 ?? 3
[
????
????
]  ??
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
)  ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
,
which is the required differential equation.
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ??
?? ?? ?? ? ???? , where ?? = fixed constant
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant
Differentiating, we get
????
????
=  ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ?
Again differentiating, we get
?? 2
?? ?? ?? 2
=  ?? 2
?? ?? ?? ?? ? ????  ?? 2
?? ?? ?? ?? ? ????
using equation (i), we get ?
?? 2
?? ?? ?? 2
=  ?? 2
??
Do yourself  2
Eliminate the arbitrary constants and obtain the differential equation satisfied by it.
(i) ?? = 2 ?? + ????
??
(ii) ?? = (
?? ?? 2
) + ????
(iii) ?? = ?? ?? 2 ?? + ?? ??  2 ?? + ??
5. SOLUTION OF DIFFERENTIAL
EQUATION :
The solution of the differential equation is a relation between the variables of the equation not
containing the derivatives, but satisfying the given differential equation (i.e., from which the given
differential equation can be derived).
Thus, the solution of
????
????
= ?? ?? could be obtained by simply integrating both sides, i.e., ?? = ?? ?? + ??
and that of,
????
????
= ???? + ?? is ?? =
????
2
2
+ ???? + ?? , where ?? is arbitrary constant.
(i) A general solution or an integral of a differential equation is a relation between the variables
(not involving the derivatives) which contains the same number of the arbitrary constants as the
order of the differential equation.
For example, a general solution of the differential equation
?? 2
?? ?? ?? 2
=  4 ?? is ?? = ???? ?? ?? ? 2 ?? + ?? ?? ?? ?? ? 2 ??
where ?? and ?? are the arbitrary constants.
(ii) Particular solution or particular integral is that solution of the differential equation which is
obtained from the general solution by assigning particular values to the arbitrary constant in the
general solution.
For example, ?? = 10 ?? ?? ?? ? 2 ?? + 5 ?? ?? ?? ? 2 ?? is a particular solution of differential equation
?? 2
?? ????
2
=  4 ?? .
Note :
(i) The general solution of a differential equation can be expressed in different (but equivalent)
forms. For example
?????? ? ??  ?????? ? ( ?? + 2 ) = ?? # ( ?? ) ?
where ?? is an arbitrary constant is the general solution of the differential equation ????
'
= ?? + 2.
The solution given by equation (i) can also be rewritten as
?????? ? (
?? ?? + 2
) = ?? ????
?? ?? + 2
= ?? ?? = ?? 1
# ( ???? ) ? ???? ? ?? = ?? 1
( ?? + 2 ) # ( ?????? ) ?
where ?? 1
= ?? ?? is another arbitrary constant. The solution (iii) can also be written as
?? + 2 = ?? 2
??
where ?? 2
= 1 / ?? 1
is another arbitrary constant.
(ii) All differential equations that we come across have unique solutions or a family of solutions.
For example, the differential equation 
????
????
 +  ??  = 0 has only the trivial solution, i.e. ?? = 0.
The differential equation 
????
????
 +  ??  + ?? = 0 , ?? > 0 has no solution.
6. ELEMENTARY TYPES OF FIRST ORDER &
FIRST DEGREE DIFFERENTIAL EQUATIONS :
(a) Separation of Variables :
Some differential equations can be solved by the method of separation of variables (or "variable
separable"). This method is only possible, if we can express the differential equation in the form
?? ( ?? ) ???? + ?? ( ?? ) ???? = 0
where ?? ( ?? ) is a function of ' ?? ' only and ?? ( ?? ) is a function of ' ?? ' only.
A general solution of this is given by,
? ?
?
?
? ?? ( ?? ) ???? + ? ?
?
?
? ?? ( ?? ) ???? = ??
where 'c' is the arbitrary constant.
Problem 6 : Solve the differential equation ????
????
????
=
1 + ?? 2
1 + ?? 2
( 1 + ?? + ?? 2
).
Solution : Differential equation can be rewritten as
????
????
????
= ( 1 + ?? 2
) ( 1 +
?? 1 + ?? 2
) ?
?? 1 + ?? 2
???? = (
1
?? +
1
1 + ?? 2
) ????
Integrating, we get
1
2
???? ? ( 1 + ?? 2
) = ?? ?? ?? + ?? ?? ??  1
? ?? + ?????? ? v 1 + ?? 2
= ???? ?? ?? ?? ??  1
? ?? .
Ans.
Problem 7: Solve the differential equation ( ?? 3
 ?? 2
?? 3
)
????
????
+ ?? 3
+ ?? 2
?? 3
= 0.
Solution: ? The given equation ( ?? 3
 ?? 2
?? 3
)
????
????
+ ?? 3
+ ?? 2
?? 3
= 0
CaseI : ?? = 0 is one solution of differential equation
CaseII : If ?? ? 0, then
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